Subjective Type

Find the shortest distance between the following pair of lines. $$\bar{r}=(\bar{i}+2\bar{j}+\bar{k})+\lambda(2\bar{i}-\bar{j}+3\bar{k})$$ & $$\bar{r}=(\bar{i}-3\bar{j}-\bar{k})+\mu (3\bar{i}+2\hat{j}-5\bar{k})$$.

Solution

$$\begin{array}{l} According\, to\, question, \\ we\, have\, two\, pair\, of\, line.......... \\ { { \bar { r } }_{ 1 } }=(\bar { i } +2\bar { j } +\bar { k } )+\lambda (2\bar { i } -\bar { j } +3\bar { k } ) \\ { { \bar { r } }_{ 2 } }=(\bar { i } -3\bar { j } -\bar { k } )+\mu (3\bar { i } +2\hat { j } -5\bar { k } ) \\ Now, \end{array}$$ $$Comparing\, the\, given\, eq{ u^{ n } }\, \, with\, the\, \, equations:{ \overrightarrow { r } _{ 1 } }={ \overrightarrow { a } _{ 1 } }+\lambda \overrightarrow { { b_{ 1 } } } \, \, \, and\, \, \overrightarrow { { r_{ 2 } } } ={ \overrightarrow { a } _{ 2 } }+\lambda \overrightarrow { { b_{ 2 } } } \, $$ $$\begin{array}{l} { \overrightarrow { a } _{ 1 } }=\widehat { i } -2\widehat { j } +\widehat { k } \\ { \overrightarrow { a } _{ 2 } }=\widehat { i } -3\widehat { j } -\widehat { k } \\ \overrightarrow { { b_{ 1 } } } =2\widehat { i } -\widehat { j } +3\widehat { k } \\ \overrightarrow { { b_{ 2 } } } =3\widehat { i } +2\widehat { j } -5\widehat { k } \\ \therefore \, \, { \overrightarrow { \, \, \, a } _{ 2 } }-{ \overrightarrow { a } _{ 1 } }=\left( { \widehat { i } -3\widehat { j } -\widehat { k } } \right) -\left( { \widehat { i } -2\widehat { j } +\widehat { k } } \right) \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =\widehat { i } -3\widehat { j } -\widehat { k } -\widehat { i } +2\widehat { j } -\widehat { k } \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =(-\widehat { j } -2\widehat { k } )-----(i) \\ Now \\ \overrightarrow { { b_{ 1 } } } \times \overrightarrow { { b_{ 2 } } } =\left| \begin{array}{l} i\widehat { \, } \, \, \, \, \, \, \widehat { j } \, \, \, \, \, \, \, \, \widehat { k } \\ 2\, \, \, \, \, -1\, \, \, \, \, \, 3 \\ 3\, \, \, \, \, \, \, \, 2\, \, \, \, \, -5 \end{array} \right| \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, =\widehat { i } \, \, (5-6)\, -\widehat { j } \, (-10-9)\, +\widehat { k } \, (4+3) \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, =-\widehat { i } +19\widehat { j } \, +7\widehat { k } -----(ii) \\ \left| { \overrightarrow { { b_{ 1 } } } \times \overrightarrow { { b_{ 2 } } } } \right| =\sqrt { { { (-1) }^{ 2 } }+{ { (19) }^{ 2 } }+{ { (7) }^{ 2 } } } \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =\sqrt { 1+361+49 } =\sqrt { 411 } =21 \\ ({ \overrightarrow { \, \, \, a } _{ 2 } }-{ \overrightarrow { a } _{ 1 } })\, \, \, (\overrightarrow { { b_{ 1 } } } \times \overrightarrow { { b_{ 2 } } } )=\, \, (-\widehat { j } -2\widehat { k } )\, \, (-\widehat { i } +19\widehat { j } \, +7\widehat { k } ) \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =1+19-14=6 \\ and\, shortest\, dis\tan ce\, between\, the\, line: \\ d=\left| { \frac { { ({ { \overrightarrow { \, \, \, a } }_{ 2 } }-{ { \overrightarrow { a } }_{ 1 } })\, \, \, (\overrightarrow { { b_{ 1 } } } \times \overrightarrow { { b_{ 2 } } } ) } }{ { \left| { \overrightarrow { { b_{ 1 } } } \times \overrightarrow { { b_{ 2 } } } } \right| } } } \right| =\frac { 6 }{ { 21 } } =\frac { 2 }{ 7 } \\ so\, \, that\, the\, \, correct\, solution\, \frac { 2 }{ 7 } \end{array}$$

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